Question

If a Sequence Is Geometric
Which sequences are geometric? Check all that apply.

–2, –4, –6, –8, –10, …
16, –8, 4, –2, 1
–15, –18, –21.6, –25.92, –31.104, …
4, 10.5, 17, 23.5, 30, …
625, 125, 25, 5, 1, …

Answer 1

A sequence is considered geometric if there is a constant ratio between successive terms. That is, each term can be obtained by multiplying the previous term by the same value (the common ratio).

Let's analyze each sequence to see if it is geometric.

1. **–2, –4, –6, –8, –10, …**
- This sequence is arithmetic, not geometric. The difference between consecutive terms is constant (-2), but the ratio between terms is not constant.

2. **16, –8, 4, –2, 1**
- To find the ratio:
- 16 to -8: \(-8 / 16 = -0.5\)
- -8 to 4: \(4 / -8 = -0.5\)
- 4 to -2: \(-2 / 4 = -0.5\)
- -2 to 1: \(1 / -2 = -0.5\)
- The ratio is constant (-0.5), so this is a geometric sequence.

3. **–15, –18, –21.6, –25.92, –31.104, …**
- To find the ratio:
- -15 to -18: \(-18 / -15 = 1.2\)
- -18 to -21.6: \(-21.6 / -18 = 1.2\)
- -21.6 to -25.92: \(-25.92 / -21.6 = 1.2\)
- -25.92 to -31.104: \(-31.104 / -25.92 = 1.2\)
- The ratio is constant (1.2), so this is a geometric sequence.

4. **4, 10.5, 17, 23.5, 30, …**
- This sequence is arithmetic, not geometric. The difference between consecutive terms is constant (6.5), but the ratio between terms is not constant.

5. **625, 125, 25, 5, 1, …**
- To find the ratio:
- 625 to 125: \(125 / 625 = 0.2\)
- 125 to 25: \(25 / 125 = 0.2\)
- 25 to 5: \(5 / 25 = 0.2\)
- 5 to 1: \(1 / 5 = 0.2\)
- The ratio is constant (0.2), so this is a geometric sequence.

**Summary of geometric sequences:**

- **Yes:** 16, –8, 4, –2, 1
- **Yes:** –15, –18, –21.6, –25.92, –31.104
- **Yes:** 625, 125, 25, 5, 1

So the geometric sequences are the second, third, and fifth sequences.